.. among other things, "imagine ELIZA passing calc ii?" (updated)

there has been a lot of discussion about the phenomenon of MOOCs (i.e. massive open online courses) but i've come across very few discussions by academic mathematicians.

there's a new opinion by k. devlin, though. in true mathematical fashion, he avoids grand-standing predictions in favor of a few valuable insights. one of them is:

At the level of the individual student, MOOCs are, quite frankly, not that great, and not at all as good as a traditional university education. This is reflected (in part) in those huge dropout rates and the low level of performance of the majority that stick it out. But in every MOOC, a relatively small percentage of students manage to make the course work to their advantage, and do well. And when that initial letter M refers not to tens of thousands but to "millions," those successes become a lot of talented individuals.

One crucial talent in particular that successful MOOC students possess is being highly self-motivated and persistent. Right now, innate talent, self-motivation, and persistence are not enough to guarantee an individual success, if she or he does not live in the right part of the wor[l]d or have access to the right resources. But with MOOCs, anyone with access to a broadband connection gets an entry ticket. The playing field may still not be level, but it's suddenly a whole lot more level than before. Level enough, in fact. And as with Google search, in education, "level enough" is level enough.

it's a good point.

one could make the argument that successful MOOC participants exhibit more persistence than on-campus students, because they wouldn't have the benefit of a community of peers to signal that they should study more or allocate time to do the work. unless they have actively looked for study partners in her/his local area, MOOC participants are on their own and their success is truly theirs ..

.. provided that the MOOC participants actually do their own work, of course.

so maybe the article title "the darwinisation of higher education" is apt;
nature also rewards those species that try their hand at camouflage.

i may be cynical, but then again, i'm also a mathematician who appreciates certainty and tries to pay attention to nonexamples.

---

my experience with web homework, though possibly unrelated, comes to mind. here was the setup:

• since the questions were multiple-choice, the grading was automatic;
• since they were allowed several chances to get each problem right,
  often without time limits and with possible access to wikis and online examples,
  most students had very good scores. many even had close-to-perfect ones.

there was always a lot of variation in their in-class quiz scores, though.

even if all of my former students were entirely ethical and did everything by themselves (in that they asked no other humans for help) there still remained a difference between what they did on a computer and what they did on their own.
..
" [thinks]
'on their own' ..
..

that betrays my biases. there's nothing wrong with looking everything up and putting it together, i suppose, especially if one always has access to the internet.

on the same token, an information engine like wοlfram α can probably do simpler but similar tasks and more efficiently, too.

many fellow educators may agree with me that most test problems in entry-level maths courses are essentially algorithmic; in fact, "good" exam problems are hard to write and i think it takes a nontrivial amount of work to write a challenging problem that requires little of either axiomatic proof or purely algorithmic computation. this suggests (but does not prove) that successors of these programs could soon perform that much better than our students on the same tasks; if you even go so far as to believe in the singularιty, then it shouldn't be that far in the future.

so as the pundits like to say, if we need new workers for an "information economy," then why should they be human workers, if machines can do a systematically better job? on a related note, imagine an eliza program passing calculus ii? that would be both very cool and very scary.

to be continued later:
some ideas on how to make money from this,
and what might go wrong.

---

added: evening, 8 jan 2013:
never mind. as i expected, my idea's not original and they already thought of it.

Coursera recently announced another route to help students earn credit for its courses — and produce revenue. The company has arranged for the American Council on Education, the umbrella group of higher education, to have subject experts assess whether several courses are worthy of transfer credits. If the experts say they are, students who successfully complete those courses could take an identity-verified proctored exam, pay a fee and get an ACE Credit transcript, a certification that 2,000 universities already accept for credit.

~ from "Students Rush to Web Classes, but Profits May Be Much Later" @nyt

indeed, if they think that students are going to (be tempted to) cheat, then they may as well make money on it ..

so i got my wish .. sort of.

oddly enough, it happens that everybody in my family is feeling ill with fever except me [1]. nobody else has any energy to do anything, except take prolonged naps.

not to be too ruthless about this turn of events, but if i'm on my own .. then it makes a perfectly good day for mathematics .. (-:



[1] first of all, no: i didn't cause this purposely. i may be itching to get some work done, but i'm not that much of a mercenary! equally odd, though, is that i've been the one who has been wearing thin sweaters and light coats and no hat while outdoors .. and running outside in 40*C^oF weather in the rain. i'm starting to believe that infectiousness [2] is either a wholly random process, or that i'm some sort of immune carrier of this particular flu .. which is still odd, since i haven't been sick lately.

[2] this is probably a made-up word. on the other hand, "infection" doesn't seem like the right word, especially as it suggests something bacterial in origin.

mildly relevant: creativity, DIY.

this is probably one of the recurring themes in my posts, such as the last one about mornings ..

.. but, if only to become more effective in one's goals, there's a lot to be said for hacking one's own habits and inclinations, conscious or otherwise. there's a difference between working effectively at a fixed task (where considerations of time can be measured quantitatively) versus being prolific at a creative task, for example.

the more i think about it, the more they seem like different modes of thinking. it's therefore appropriate to consider ways of making these experiences more efficient and separately so ..!

---

Creativity in the Wild: Improving Creative Reasoning through Immersion in Natural Settings

Ruth Ann Atchley¹, David L. Strayer^2*, Paul Atchley¹

1 Department of Psychology, University of Kansas, Lawrence, Kansas, United States of America, 2 Department of Psychology, University of Utah, Salt Lake City, Utah, United States of America

... However, the impact of nature on higher-level tasks such as creative problem solving has not been explored. Here we show that four days of immersion in nature, and the corresponding disconnection from multi-media and technology, increases performance on a creativity, problem-solving task by a full 50% in a group of naive hikers. Our results demonstrate that there is a cognitive advantage to be realized if we spend time immersed in a natural setting. .

---

Creativity Happens When You Least Expect It

By Sian Beilock | Dec 07, 2012

... But not all tasks require working memory for success. In fact, sometimes people’s ability to think about information in new and unusual ways can actually be hampered when they wield too much brainpower. This means that what we think of as our optimal time of day, may not be optimal for everything.

Recent research confirms this idea. In a paper published last year in the journal Thinking and Reasoning, psychologist Mareike Wieth and her colleagues found that when people have to solve “insight problems” that require a high degree of creativity, solvers are much more successful when they tackle these problems at the time of day in which they are least alert.

---

i must say: the more i learn about the human brain, the more amazed i am at how it works ..!

a quick one, while he's away .. (updated)

// started: 24 dec 2012, mid-morning.

i haven't worked in a coffeehouse in a while:
comforting but slightly unfamiliar.

the background christmas music is distracting, though.

// added: 25 dec 2012, early morning.

due to oceanic jet lag, i woke up dreadfully early, well before sunrise on EST .. and managed to improve a technical lemma in one of my manuscripts.

so if all goes well, then this will make an extant proof even clearer,
and will cut away ~2 pages from the previous writeup.

not bad: a little thinking done, even before any of the family's woken up. you see, the older i get, the harder it seems to carry a normal conversation, first thing in the morning.

put otherwise, i feel more encouraged to work out ideas, right away upon waking up ..

.. as if it were a perfect time to hack my unconscious,
see what strange intuitions have occurred to me while i slept, the night before ..

.. so it happens fairly often that i can't wait to get up and start the day, even if i'm not fully awake. ultimately, i'm a more effective (and less irritable) person when left alone in the mornings .. at least long enough to sort out an idea or two.

// added: 25 dec 2012, late afternoon.

i was about to check the arXiv .. but then realised:

if i do look for preprints, then it's likely that i'll browse through some of them and start thinking about ideas. once that happens, then the whole week will turn into something degenerate.

it won't really be a holiday, nor will it be a proper workweek. i'll probably judge it as one or the other; in either case, i won't be satisfied by what will happen.

no good can come out of this.

so i left the arXiv alone, for now. it's one thing to work in the early mornings when everyone else is asleep .. but if i've made a point to visit family during the holidays, then shouldn't i be consistent and set aside the maths?

[sighs]

mildly relevant: campus life is relatively rare.

this is an excerpt from an article called "the end of the university," as found on the american interest.

more appropriately, it should be called "the end of the american university" because there is a strong focus on tuition, student debt, and the 'business' of running a university.

to wit: quite a few places in europe still have very affordable [1] university education for all citizens of the e.u. without a lot of tuition fees .. though the tide seems to be turning, especially in the u.k.

it's funny .. in the united states, they're talking about tuition-free higher education, whereas much of europe is discussing the exact opposite!

anyway, about that article ..

-✂--
..
To borrow an analogy from the music industry, universities have previously sold education in an “album” package—the four-year bachelor’s degree in a certain major, usually coupled with a core curriculum. The trend for the future will be more compact, targeted educational certificates and credits, which students will be able to pick and choose from to create their own academic portfolios. Take a math class from MIT, an engineering class from Purdue, perhaps with a course in environmental law from Yale..
..
-✂--
..
It’s worth noting that while the four-year residential experience is what many of us picture when we think of “college”, the residential college experience has already become an experience only a minority of the nation’s students enjoy. Adult returning students now make up a large mass of those attending university. Non-traditional students make up 40 percent of all college students. Together with commuting students, or others taking classes online, they show that the traditional residential college experience is something many students either can’t afford or don't want. The for-profit colleges, which often cater to working adult students with a combination of night and weekend classes and online coursework, have tapped into the massive demand for practical and customized education. It’s a sign of what is to come.
..
-✂--

[1] well, compared to private universities in the u.s., at least.  from what little i've read, the enrollment cost to a university student per semester is less than 1000€ per year in spain and france, and i think that finnish universities don't charge tuition at all ..?

it's not the fault of the holidays .. it's just me.

this and last week i've lacked focus. every day i get up, i try to work, but little if anything happens. there are some ideas on a few sheets of paper from the last few mornings ..

.. what's not made it to the recycling bin, anyway ..

.. and then there are a few manuscripts to rewrite, papers that i told myself i'd actually read [1], co-authors to whom i should really write back ..

the only thing that's not on my to-do list is preparing another talk, which is relieving. it's not that i don't like giving talks; i could use a break from the road, though.

---

i'd like to think that i'm still tired from recent travels .. which i probably am, but that's not entirely it [2]. it really is a lack of focus, or more accurately: commitment.

lately i've found it difficult to choose what to do. i have too much on my plate these days ..

barely relevant, but ..

after reading this mentalfloss article,  now i'm tempted to use the character/symbol ..

(literally, "that" .. a combination of thorn and t.)

.. as a replacement for "so/such that" ..! (-:

mildly relevant: disruption in education.

an interesting call to action,
as passed to me in november by a colleague.

(FYI: here "disruptive" is meant in the sense of 'disruptive innovation.')

But you know what? Those classes weren’t like jazz compositions. They didn’t create genuine intellectual community. They didn’t even create ersatz intellectual community. They were just great lectures: we showed up, we listened, we took notes, and we left, ready to discuss what we’d heard in smaller sections.
-- ✂ --
Outside the elite institutions, though, the other 75% of students—over 13 million of them—are enrolled in the four thousand institutions you haven’t heard of: Abraham Baldwin Agricultural College. Bridgerland Applied Technology College. The Laboratory Institute of Merchandising. When we talk about college education in the US, these institutions are usually left out of the conversation, but Clayton State educates as many undergraduates as Harvard. Saint Leo educates twice as many. City College of San Francisco enrolls as many as the entire Ivy League combined. These are where most students are, and their experience is what college education is mostly like.
-- ✂ --
We have several advantages over the recording industry, of course. We are decentralized and mostly non-profit. We employ lots of smart people. We have previous examples to learn from, and our core competence is learning from the past. And armed with these advantages, we’re probably going to screw this up as badly as the music people did..

(continued in "napster, udacity, and the academy" @clayshirky)

don't worry: i still remember .. and i understand completely.

less than half of december remains. i guess it means the end of the year is truly upon us .. and for many of you, the season of job applications is partially over [1].

of course, the pessimist in me would point out the many deadlines yet to come .. and how there is really no rest for temporarily-hired persons in maths.

what a life!

after grading batches and batches of final exams,
doling out grades for the course,
answering upset student emails [2],
maybe a free afternoon to spend revisiting an idea,
getting back to co-authors ..

.. after that, it then becomes time to finish another 30-50 job applications [3]!

---

so to those of you on the market, this is to say: i'm sorry. yes, it's not my fault, so it's not really an apology .. but i know your pain, have felt it, and you know how i feel about it too.

i remember how it feels to worry so much about the future that you cannot even concentrate on what really matters to a mathematician ..

.. that matter, being mathematics, the freedom of calm and contemplation ..

.. so maybe this is my way of dealing with a mathematical version of survivor guilt. at any rate,

mildly relevant: a web-based, collaborative latex editor.

interesting: writelatex is a latex version of etherpad, a collaborative writing and editing tool.

i would imaging that coupling dropbox with your favorite $\LaTeX$ program would accomplish something similar but without the option of editing the same document simultaneously.

(to me, though, the only instance where this could be really useful would be if several authors are composing the introduction of a paper together.)

for mathematicians (like me), an open problem is like a revealing mystery.

it's for title/abstracts like these that i constantly check cνgmt for updates. (sure, some researchers do post survey articles and lecture notes on the arχiv, but not as often.)

L. Ambrοsio - M. Colοmbo - S. Di Marinο

Sobοlev spaces in metric measure spaces: reflexιvity and lower semicοntinuity of slοpe

Submitted Paper (2012)
Keywords: Sobolev spaces, Metrιc measure sρaces, Weak Gradιents

Abstract. In this paper we make a survey of some recent developments of the theory of Sοbolev spaces $W^{1,q}(X,d,m)$, $1 < q < \infty$ in metric measure spaces $(X,d,m)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheegεr's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $m$. We also discuss the lower semicοntinuity of the slope of Lipschitζ functions and some open problems.

one cool thing about these kinds of expositions is that open problems of the field are explicitly stated, just put out there. it's not that i expect to solve them, but there's something .. enchanting? alluring, i suppose, about encountering something that nobody knows how to solve (yet).

I love rumors! Facts can be so misleading, where rumors, true or false, are often revealing.
~ col. hans landa

it drives one's ideas, sharpens one's focus to some good end;
also, open problems suggest ..

.. though mathematicians vary by talent, inclination, and drive in very large degrees ..

.. that we are all equal in a few ways, at least until someone solves the problem at hand. then again, there are always problems and unknowns, just like there are always books i've never read in any public library.

their existence is somehow very comforting to me. (-:

on math ed: the (possible) danger of breaking down problems ..

whoa -- from "u.s. math achievement: how bad is it" @psychologytoday

When the researchers broke problem-solving activities down into procedural activities and conceptual activities, they expected to find that the higher performing countries engaged in more conceptual problem solving. They found no such difference. But then they took a second step. They coded the data based on whether the teachers made the conceptual problems easier by converting them, for the students, in to procedural problems.

Looked at this way, it became clear that the US was an outlier (as was Australia, the only other low-performing nation in the study). Teachers in the US almost always converted challenging conceptual problems into procedural problems. In doing so, they did exactly the wrong thing. According to a seminal study by Hiebert and Grouws (2007) the two features of instruction that predict good math outcomes are

1. Being explicit about the conceptual structure, and interconnectedness, of mathematics
2. Allowing students to struggle to understand mathematical concepts.

By converting conceptual struggle into procedural learning, US math teaches were unintentionally depriving their students of two crucial elements of effective learning.

the unnerving thing is that, to me, "breaking down a problem" seems like a natural thing to do. it's how research goes, all the time.

the point, however, seems not for the teacher to do all of the reduction .. as (s)he would already know how .. but for the students to learn how to do it.

.. there and back.

today and yesterday i felt like everyone treated me a little like a carton of milk, on which the expiration date was hard to read.

they: "janus, hi! how long are you in town?"
me: "hi! er, well .."

.. it was a short visit this time: a little more than 48 hours. i felt like i was constantly saying hello and sorry and measurablε dιfferentiable structure and goodbye to many colleagues, both new and familiar.

it happened that my colleague/friend/host asked me if i've visited his (new) university before. i nodded.

thrice before, in the last 10 years:
one of the first conferences that i ever attended, too.

speaking of a next time, another colleague asked me if i'd be free to visit next spring .. which means, i hope, late spring. honestly, i need a break from traveling [1].

besides that, there's something about the finnish winter that makes me want to retreat indoors, and not come out for anything but trail running, indoor rock climbing, a long session of sauna, or a pub crawl.



.. on an unrelated note, there's something very soothing about traveling on trains.

---

the more i think about it, the more collaborations feel unnatural to me. i find them stressful, honestly .. though it's often a good type of stress that leads to some productive end.

one problem is that i'm just not that quick .. not right now, anyway;

if i believe it and if i act effectively towards it,
then who knows: maybe i can actually become smarter.

on the other hand, maybe i should just let myself be "stupid" and throw out ideas, instead of sitting quietly, contemplating, working them out into a polished form by myself.

to a certain extent, it doesn't matter if a given idea is a good one or a bad one. what matters instead is that you eventually get a few good ideas.

so the point is to generate good-enough ideas and resolve them effectively, and more processing power (in terms of number of minds thinking about it) generally leads to swifter resolutions.

still, i hate sounding stupid and i can't stand not knowing what to do next .. which is a frustratingly impasse, i tell you. maybe i just have to get over it.





[1] in 2012 i've been to lappeenranta, madrid, new york, tampa, ann arbor, lansing/cadillac, turku, amsterdam & brussels & brugge & lille & reims & strasbourg & heidelberg & frankfurt, krakow, jyvaskyla, new york again, pittsburgh, paris, san francisco, london, madrid again, segovia, and jyvaskyla again .. which totals 88 days (or ~3 months) i spent out of town ..

[realises]

.. ye gods! now that i think about it, i criss-crossed the atlantic four times this year, and gave fourteen (14) talks & lectures. is something wrong with me? i must be a masochist..!

mildly relevant: story-telling, neurologically speaking ..

interesting.. if the brain really works out this way, then this could prove useful for more effective teaching or perhaps giving better talks:

Evolution has wired our brains for storytelling—how to make use of it

Now all this is interesting. We know that we can activate our brains better if we listen to stories. The still unanswered question is: Why is that? Why does the format of a story, where events unfold one after the other, have such a profound impact on our learning?

The simple answer is this: We are wired that way. A story, if broken down into the simplest form, is a connection of cause and effect And that is exactly how we think. We think in narratives all day long, no matter if it is about buying groceries, whether we think about work or our spouse at home. We make up (short) stories in our heads for every action and conversation. In fact, Jeremy Hsu found [that] "personal stories and gossip make up 65% of our conversations."

~ from "the science of storytelling" @lifehacker.

lessons in (not) traveling.

[written earlier on the train]

i don't know if i'm tired from work or tired from traveling.

as for this week, i'm taking a short trip and spending a few days at a nearby university: a colleague (and friend) moved there for a position and i've been invited to speak in his new seminar, maybe work together on a problem or two.

---

it's been a week since i've returned from a 2-week stay in spain. i don't know where those seven days went, but they didn't amount to too much. then again, it's hard to compare the daily grind to, say, working for hours with collaborators every day for a while; the latter case certainly sets a rather high standard, of course.

i know that i showed up to the office, committed to a list of tasks, and completed some of them .. yet it feels like i've not accomplished anything since coming back. part of the problem is that i've been too preoccupied with this next trip, for no other reason than i have to plan for it and stick to the plan.

it is a curse to have a one-track mind, incapable of multi-tasking.

at any rate, a week is too short: i should have arranged more space between these trips .. at least two weeks, in order to rebuild a routine.

habit and routine have become my main tools to develop any kind of work efficiency.
it sounds boring, but it really works.

i suspect that this is something that everyone just knows ..
.. well, except me, who had to actually learn it.

the holidays are coming. taking time off sounds very good right now, but there are so many things to do. the year is ending and all i can think about are the things i've not yet done but wanted to do.

those kinds of thoughts are exhausting ones, and they only lead to a vicious cycle of unproductivity and self-recrimination ..

.. that is, i think about my shortcomings,
feeling bad about them, i sulk and do nothing;

having done nothing, i develop new shortcomings,
then think about them again .. 7-:

i don't know if i'm really tired, but that's how i feel.



[written later, having arrived]

epilogue: it's always worth visiting friends. (-:

mildly relevant: the fault lies not in the stars ..?

it may seem like students are getting worse every year .. which is what an educator would find very convenient to believe. it would imply that we're doing a good job but everyone else isn't.

i find that answer a little too convenient.

it is true that school standards change with time .. but i remember suggesting once to colleagues that the reality may be purely relative to our perspective. instead, what if we are simply getting more familiar with teaching the same topics?

But if you've never been to a place before, you need more than a description of a place; you need an exact definition, or a precise formula for finding it. The curse of knowledge is the reason why, when I had to search for a friend's tent in a field, their advice of "it's the blue one" seemed perfectly sensible to them and was completely useless for me, as I stood there staring blankly at hundreds of blue tents.

This same quirk is why teaching is so difficult to do well. Once you are familiar with a topic it is very hard to understand what someone who isn't familiar with it needs to know. The curse of knowledge isn't a surprising flaw in our mental machinery – really it is just a side effect of our basic alienation from each other. We all have different thoughts and beliefs, and we have no special access to each other's minds..

~ from "why we are so bad with directions" @bbc

life wasn't simpler, back then.

over the weekend i learned about blogger's label feature ..

for an example, click here; also .. [0]

.. so i ended up reading quite a few posts i wrote when i was in the middle of my ph.d. and shortly after i started doing actual research. it makes for more interesting reading than my life now, but .. ye gods:

I. it was a dark, uncertain point in my life. i forgot how often i worried about whether everything would work out or even whether i was cut out to be a mathematician [1], how it took so long to ever learn anything .. much less work with it .. and then there were 2-3 rounds of job searches: one just before the financial crisis hit with full force, the other(s) after it.

in comparison, my life now feels relatively secure and even routine.

that's not to say that i've stopped worrying. rather, i think i worry just as much as i used to [2] .. just that i've gotten used to not getting any resolution out of it, and gotten better at pushing it out of mind.

II. i wrote a lot more often about technical details, and i don't exactly know why. maybe it was because all this research stuff was new to me and i found it hard to separate it from the rest of my life.

maybe it's because by writing them, i could partially convince myself that i had some idea of how they worked .. which i didn't, of course.

maybe it's because i've encountered enough non-experts, relative to my field, and have given enough unsuccessful explanations that i've grown tired of writing or thinking about them, during my non-work hours.

contrary to popular expectation, not all of us researchers work all the time;
these days i would consider myself a non-example.

who knows? maybe it's progress .. that i see a bigger picture now and that the details, though still essential and important, aren't the main focus anymore.

related to this, my research problems have been asking why a bit more often than how, these days.

III. travel wasn't routine to me, yet. to be fair, i didn't particularly like traveling back then, either, but i was more excited by the trips i took.

the first flight to finland (2003) was an adventure to me, for example.

quotable: not a 'can say,' but "can do" attitude.

from "the second test" @randsinrepose:

I am not suggesting that the hardworking people in these other disciplines don’t have amazingly complex and difficult jobs, but I do think they should be able to clearly describe the work they do and the value they provide… to anyone. They need to pass the Second Test, and that means being able to fully and clearly explain your job to the rest of your team not with words, but with action.

Most folks believe that if they can describe a job that they can do it. Most folks are wrong. I’ve been spun and burned by too many fast-talking, charismatic experts in my career to trust anything but results. The Second Test is not the exclusive domain of engineers. In most groups of people, there is a means by which you earn your stripes. The difference with engineers is a combination of their low tolerance for spin and their deep desire for measurability.

.. and in a similar vein, from "surely you're joking, mr. feynman!"

All the time you're saying to yourself, 'I could do that, but I won't,'--which is just another way of saying that you can't.

class dismissed: as for the rest ..

overall it's been a good visit, but intense and tiring. i've tried to do a good job with the lectures, but not at the expense of research.

.. which would explain the suboptimal planning of my lectures!

as of now, my colleagues and i have gotten headway into a well-defined project, maybe two.  there are still many details to consider, but we've thought long enough and hard enough on a few problems that i believe something can be done with them.

one part makes me cringe, though: i might have to revisit some of the basics of some slightly arcane topics in functional analysis ..

.. such as the weak-star operator topology ..! [1] 7-:

---

to be fair, i think that it's been easier on me than on them.  often i do most of the listening and little of the talking.  it went the other way around, this time.

over the last two weeks i felt like i explained a lot about a theory that nobody seems to study very much .. which is a shame, since i've consistently found it a rather useful theory. by sheer exposure and effort, i think i've also convinced my collaborators of the same.

so i did end up being useful, in some fashion .. which is relieving.  in spain these days, research funds aren't easy to come by.  imagine if they had invited me and gotten little out of the deal! (-:

for my own part, i learned quite a bit about fractals .. and in some sense, fubini's theorem, too.



[1] to be precise, the wiki directs to the weak operator topology, not the weak-star operator topology .. but on the other hand, what i would call the weak-star operator topology differs from what the corresponding wiki would suggest; i claim that one doesn't actually need the underlying source and target spaces to be the same .. much less hilbert spaces .. \-:

class in session, part 3: teacher (not student!) evaluations.

so today was my final lecture;
tomorrow will be my last day in spain .. for a while, anyway.

---

part of me wonders whether it was really wise to agree to a lecture series.  i don't know if the students who attended got much out of my ramblings about metric spaces, geometry, and differentiability.

originally i thought that it would be best to focus on the ideas and how the proofs are structured [1], so a lot of what i presented consisted of special cases of theorems and sketches of proofs.  looking back at it now, i don't think i left enough of a trail to follow what was going on .. and i admit that i went rather fast [2].

i skipped a lot, yet still didn't get very far;
in the end, maybe the compromise was for nothing.

i don't know.
they're over now; that's all i really know.

my colleagues tell me that they really enjoyed the course .. but from experience, if an expert thinks your course is interesting, then you have to ask: is this really good for the students, to whom the course was directed? [3]

the course shouldn't be about the educator who gives the lectures, and it shouldn't be about impressing colleagues either. it should be about the students who are to learn from the lectures.

that's the whole point of a lecture, after all.



[1] put another way, details are easily accessible, by way of the full proofs in the referenced articles.  ideas and intuition --- the kinds of things that only an expert can convey quickly and efficiently --- are harder to come by.

[2] that's my fault, really. i promised more topics than i could really discuss in depth, despite the fact that there were four lectures.  i underestimated how long it takes to motivate the topic (1 hr) and what is really required to prove standard lemmas that i use every day (1 hr).  in the remaining time i discussed two topics, each of which would be a full finnish seminar's treatment (2 x 2 = 4 hrs).

[3] i think it would be fun, one day, to teach a semester-long advanced topics course, but i realise now that it comes with a personal price.  if you teach nothing but advanced topics, then it disconnects you from the day-to-day reality of students, their experience, and their expectations.

there's a school of thought that says that "if the student is strong, then (s)he will rise to the challenge" .. but that's like saying that mathematics is only for the masochists .. which some would believe to be true, too.   i believe otherwise, that if the student is inspired and if (s)he is shown that (s)he can succeed, then (s)he will succeed.

mildly relevant: startup $\perp$ research?

Beware of research. If an undergrad writes something all his friends start using, it's quite likely to represent a good startup idea. Whereas a PhD dissertation is extremely unlikely to. For some reason, the more a project has to count as research, the less likely it is to be something that could be turned into a startup. I think the reason is that the subset of ideas that count as research is so narrow that it's unlikely that a project that satisfied that constraint would also satisfy the orthogonal constraint of solving users' problems.

~ from "how to get startup ideas" by paul graham.

well, in that case i'm set:
being a researcher, that means it's easy to leave the money-making to the mercenaries.

---

maybe i'm channeling cayley and hardy overmuch [1], but there is something relieving in doing work that is not immediately applicable to anything. for instance,

• you're not arming militaries with new, more dangerous weapons,
• you're not making poor people poorer and rich people richer,
• you're not coding an iphone app that just makes it more likely for tech-addicts to further ignore each other at the dinner table.

to do useless work is to do no harm; i can get behind that!


[1] “I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications.”

class in session, part 2: third down, incomplete pass .. (also: #1200)

so i punted.

today was already lecture 3 of 4 and i planned sufficiently poorly so that the $\LaTeX$/PDF notes for lecture 2 were still only half-complete.  despite this, they remain self-contained, readable, and just-barely-suitable for public consumption [1].

this pains me nontrivially [2].

if these lectures were part of a "real" course .. that is, where one would solve problem sets and get actual credit .. then the students would be lost.

this is not an exaggeration.  i can tell something isn't quite clear, if only because my collaborator was part of the audience, and she had a great many questions about some points i made.

so if she, an established researcher, could not catch everything then what chances would a ph.d. student have to be able to catch something .. especially if this is not their field of interest?

[sighs]
so yes, i gave up.

between a final push of research collaboration this week and writing up my own notes [3] for lecture 4, i see little-to-no time available for catching up with the $\LaTeX$ for lecture 2 and proceeding with lecture 3 from this afternoon.  so if is infeasible to do so, then why bother?

instead i posted onto my webpage some PDF scans of some notes from previous talks.  it's not a perfect solution, but it's better than nothing.  more than that, it's important to make something available for those students who may actually want to look at the details [4].

yes, probably none of the students will actually do this .. but if there is a nonzero probability that one might, then it is worth doing.

i also have other reasons for disappointment.  for example, i really wanted to have the lecture notes in wiki format.  my reasons involve symmetry:

on research and moving targets.

this week starts the second week of my visit to spain.

i think it's going well. we're discussing a lot of problems, setting up projects, seeing which approaches make sense and which could be wishful thinking. two weeks isn't forever, you see;

it's good to plan what can be done later, on our own.

---

regularly my colleagues have been asking me specific questions, to which i don't have any answers. this is completely natural, of course ..

.. it's not like they know the answer either,
.. but i still can't stand not knowing.

it reminds me of the film butch cassιdy and the sundance kid:

there's this one scene where the two main characters are practicing their shooting, and sundance is missing every shot.

when asked how he can hit anything, sundance starts walking to the doorway and simultaneously shooting .. only to hit the bullseye every time.

"I'm better when I move," he says.

research-wise, i don't think i've ever been able to solve an explicit problem when "standing still" .. that is, in one concentrated session.

(this includes dedicating a full 6 weeks to it and working on nothing else.)

---

it almost always occurs by accident.

sometimes it's when i'm doing something else.
other times it's some time later after i've finally give up (usually at least a week).

either way it just dawns on me that two separate topics are related in exactly the way that i need them to relate .. at which point, it's not that i feel particularly happy that i hit the target. there's a base amount of amazement that it actually fits together, sure, but most of all i feel stupid that it took me that long!

that's not exactly how it works, though: it probably takes a while to absorb all the information, how everything connects logically, and to step sufficiently far away from it all in order to see the truly essential parts.

i guess seeing something "easy" after months of fruitless activity is just a little disarming to me ..

.. then again, even after the answer presents itself, there is usually a lot of technical work before the proof is fully rigorous.

so usually after the writeup, i feel slightly better .. that yes, it does take more than a lucky break to prove theorems.

if the result is worthwhile, then usually it requires quite a bit of hard work .. \-:

computers have a "shutdown" button .. but if brains were only that easy.

i wouldn't say that i'm tired. my mind is active, even after having worked for most of the day [1] but focus is hard. despite it being a bad idea for many reasons [2], i feel like working.

lately it's becoming hard to "shut off" at night.

there are always ideas that come to mind. of course, most of them don't work and some of them are (admittedly) crazy .. but getting a new idea is addictive. moreover, trying them out is like gambling:

one doesn't expect to win anything;
then again, if one does happen to win .. (-:



[1] woke up at 7:30am, wrote up a final set of lecture notes over breakfast, thought about a random problem or two before heading to my guest office at 9:30, answered some emails and ignored others, met collaborators and discussed ideas and technicalities for projects until the time for lecture, gave a 1-hour lecture, back to collaboration, late Spanish lunch at 14:00, back to work for a few hours ..

[2] until recently i had been consistently good with not doing maths until going to bed. at some point i developed a kind of mental allergy: now i can't sleep right after thinking about technical details.

class in session, part 1: false start.

---

i don't know why, but after the lecture i felt a tremendous sense of despair, as if i had done something very wrong. it's not like it was a complete disaster, and as a first lecture i guess it started very reasonably ..

.. and maybe that's it; maybe i didn't plan out the lessons well enough. being so used to the 2-hours afforded by finnish lectures and seminars now, i'm quickly running out of time.

right now i don't know how to fit in all the topics. it wouldn't be as much of an issue .. had the talks not been advertised as a short seminar course.

i don't know where the time went. maybe in principle it was good to motivate the main results .. but somehow i found myself 40 minutes into the lecture and i hadn't even defined the basic object of the "course."

[sighs]

there's a lot of work to do with the material .. and on top of that, i'm really here for a research collaboration. i guess being away from finland for a little while doesn't quite change too much: there's always so much work to do, and so little time.

---

preparing for class, part 3: lecture notes.

that monograph thing that i mentioned earlier is just an idea, really;
i don't know if i should be the one who would write such a thing.

to clarify, it's not that i don't think a monograph should exist on the subject [2]. it's just that someone more senior and established in the field should do it [1] .. or rather, someone more trustworthy and with a better view of the Big Picture, both in regards to the field as to how it started and how it is now.

anyway, i don't trust myself to do it .. that is, to do a good job with it.

the last time i tried to write an exposition, it was about the p-Laplace equation and the first draft was such a mess that i despaired of it ever seeing print.

most of it is a re-telling of DιBenedetto's story, of course, but with an emphasis on techniques that can easily be extended to both non-smooth spaces and functions on them. for those of you who know the field, you probably know there are enough of these kinds of expositions so that a better one than mine exists and is accessible.
..
..

.. maybe i'm being too rigid in my outlook, though.

---

thinking about it, the world's gone sufficiently digital as to allow a continuum of document types. on one end, we have well-polished textbooks and peer-reviewed research articles .. but there are less formal ways, too.

• take, for example, this prιmer on differentιable structures on metrιc spaces. maybe it will see print in a journal, maybe not (though i hope it will), but as it exists now, i think it's a great resource. it serves the very useful purpose of a clear, concise exposition without suffering the troubles of formal publication.
• a similar resource is a technical report that the advisor wrote, years ago: i still favor it over the standard GMT references, when thinking about fΙat currents. the topics are based on .. come to think of it, a series of lectures he gave one august.

a final decision can wait, i guess. the publication of monographs seems such a formal affair to consider. i think i'll start small. i've already developed the practice of sharing my handwrit talk notes with audience participants, and occasionally posting scans of them on my homepage.

being that the contents of these lectures will be a little substantial, maybe it will be good to have them in a digital, searchable format and easily accessible on the web ..

.. so it sounds like a wiki is in order!





[1] then again, colleagues of mine have co-written a book about some topics on metric space geometry. one of them defended his ph.d. the same year as i did .. so if he can do it, then maybe i can too?

[2] to be honest, i think there are too many books and papers out there. this is related to job candidates essentially being required to publish a lot of papers .. and with teaching loads as they are, these days, odds are that not every paper will be a strong one. maybe it's not a bad idea to show restraint and buck the culture .. because if we don't, then this is only going to get worse!

article post: not running away from a problem ..?

from "uncertainty, innovation, and the alchemy of fear"
by jonathan fields (@ the99%), found on 13 may 2012.

-- ✂ -- --

.. "People who tolerate ambiguity may be able to work effectively on a larger set of stimuli or situations, including ambiguous ones, whereas intolerant individuals will avoid or quickly stop treating such information.”

Problem is, with rare exception, when faced with the need to live in the question, most people, creators included, experience anything from unease to abject fear and paralyzing anxiety. And there’s a neuroscience basis. According to fMRI studies, acting in the face of uncertainty lights up a part of the brain known as the amygdala, which is a primary seat of fear and anxiety. That sends a surge of chemicals through our bodies that makes us want to run.

-- ✂ -- --

interesting choice of words: "living in the question," as if the problem causes you to live in your own little world .. which does agree with my experience, admittedly.

quite a few times, i had to convince myself that i should just try something .. no matter how stupid: just the first idea that comes to mind, anything. if by chance it works, then the problem stops: great! if it fails, however, then typically it does so for a very good reason, so if i understand why, then the next attempt will be more likely to work, and so on ..

as they say, sometimes solving the problem doesn't matter so much as learning about the problem, and finding interesting things.

preparing for class, part 2: preparedness.

likely i'm deeply biased about mathematical culture and conventions, but i always thought that lectures and mini-courses belong to more senior and experienced people.

for instance, i recall now some lectures that colleagues gave in chennai, a few years ago, as well as an earlier summer school in barcelona. these were very fine lectures indeed;

in particular, i admired their self-restraint.

it was clear that the lecturers knew many things, but opted to discuss a handful of ideas with care, clarity, and motivation. there was no real grand-standing and strutting: the ideas were the stars of the show.

having been impressed by this approach, i think it best to follow the same traditions. the ideas should come first.

i already sent a title/abstract of these lectures. thinking about it, many of these ideas are very new .. new as in the last few years, so this is going to require a great deal of care. i don't have too many references to depend on, and those in the literature require ..

.. some experience to decipher, you could say. so unless i expand the topics further ..

.. which runs the risk of cramming too much into the space of a few hours [1] ..

.. then whatever i present will run the alternative risk of becoming a standard reference for these topics [2] .. especially as i'm thinking of assembling the lecture materials into some kind of a monograph ..!





[1] i've gotten a little too used to the length of finnish seminars, which is 2 x 45min. when you think about it, that harrowing week where i gave two seminar talks would contain the same content as 4 usual lectures elsewhere .. and in case you were wondering, yes: that week was spent beta-testing some preliminary lecture material for next month.

if you're surprised, then you shouldn't be. come on: do you really think i'm that un-lazy? (-;

[2] re-reading what i just wrote, it sounds arrogant and self-serving, but i don't know any other way to describe it. in an expositional article, the advisor once spent a chapter outlining one particular topic, but that was 5-6 years ago and due to brevity, it doesn't discuss how exactly one uses the theory. there are some new tools available now, too, which warrant their own exposition.

preparing for class, part 1: first day jitters (to come).

so i'm scheduled to give a few lectures this november, in spain.

originally in the invitation, i was asked to give a talk or two .. where two seemed like an important detail to consider. so i asked exactly how many talks they wanted and what topics the audience would find familiar. the organiser responded, then i wrote back, then he did ..

.. and one thing leads to another ..

.. so rather than rushing in my usual manner through highly technical details, i'll instead go through a half-expository route .. and probably promote my own (recent) results as well.

i think this to be a great honor;
that said, i'm deeply worried about it.

mildly relevant: i am a masochist .. and likely, so are you.

this headline's been floating around the web recently. in particular, the excerpts below are explicitly from an article @arstechnica:

as the authors of a new paper on math anxiety point out, most forms of higher math didn't even exist until a few centuries ago. It's very unlikely that this sort of anxiety has evolved a specialized brain structure dedicated to it. So, the researchers used a combination of math quizzes and functional MRI scans to identify the areas of the brain associated with the fear of math. It turned out to be one that was previously associated with the experience of physical pain. And it doesn't appear to be the first time that area has been borrowed for other purposes by evolution: it also helps register the discomfort of social rejection..

.. well: that will certainly increase the number of mathematics majors! 7-:

strictly speaking, either we are masochists,
or that we've just had bad teachers that weren't shy about calling us stupid ..

.. others have indicated it can be triggered by indirect forms of pain, such as social rejection. But the authors note the majority of published studies associate it with pain, and that it's possible to induce the experience of pain simply by stimulating the insula ..

okay.  fine, so we're masochists;
my parents could have told me as much!

it can also be a point of pride, i suppose: mental toughness, in face of adversity and perceived pain. actually, it has a stark romanticist ring to it .. (-;

actually, there's more:

 their conclusion is that we are actually dealing with a pain response, and one that's not triggered by doing math. "It is not that math itself hurts; rather, the anticipation of math is painful," they suggest.

to be fair, what else triggers the pain center of the brain? does physics do the same thing? literature? what about sports, where there may be actual pain involved?
..
..
..
.. now that i think about it, wouldn't the readings change if the human subject had nothing to fear from maths? after all, the study is on math anxiety, not biologically-manifested cognition of mathematics.

i must be jumping to conclusions, asking the wrong questions. the point isn't whether maths inherently triggers pain. rather, if maths triggers anxiety, then does it manifest in the form of pain?

a day in the life: impromptu talks.

i think i'm developing a reputation for responsibility, punctuality, and good cheer ..

.. which, i fear, could very well lead to disaster!

the last thing i need is one more goal to juggle and meet,
and the potential fallout from if/when i drop the ball with something important.

maybe i should start breaking a few promises and failing to show up for a meeting or two .. you know, small things to keep my colleagues on their toes?

on a related note, i've been having a lot of meetings lately .. but more on that later.

---

so yesterday i addressed the department .. or at least a large chunk of it. of course, i didn't intend on this at all and had i known, i'd have most likely refused.

it all began when

mildly interesting: a look back.

sometimes i forget how history unfolds .. even within the recent history of american mathematics:

"And it's kind of interesting to see what happened to engineering. So like when I got to MIT, it was 1950s, this was an engineering school. There was a very good math department, physics department, but they were service departments. They were teaching the engineers tricks they could use. The electrical engineering department, you learned how to build a circuit. Well if you went to MIT in the 1960s, or now, it's completely different. No matter what engineering field you're in, you learn the same basic science and mathematics. And then maybe you learn a little bit about how to apply it. But that's a very different approach. And it resulted maybe from the fact that really for the first time in history, the basic sciences, like physics, had something really to tell engineers."

~ from "noam chomsky: on where a.i. went wrong" @theatlantic

wow: a lot changes in 50 years. if i had to name one of the top mathematics department in the world now, MIT would be up there ..

.. then again, maybe i'm making too much of this personal anecdote. after all, how many top maths departments were in the united states before the 60's, anyway?

a day in the life: lack of willpower, and footnoted opinions about on marginal utilities of productivity and on collaboration.

(this was originally written after sundown on sunday, 4 november 2012.)

the sentiments are pretty common in my life, so this post is probably an instance of the usual affect of academic researchers who work all the time. lately i've developed the opinion that this affect mightn't be the most effective means of working, especially if one trades in the coin that is creativity and innovation [1].

with that in mind, the short post below is about trying to hack my own personal productivity function, by way of a morning routine that exploits the temporary "reboot" effect after a good night's sleep.

despite the brevity of the post, i surprised even myself by how many side opinions i have in this business. some are about the nature of collaboration .. which i'll write more about in a future post.

---

---

---

in principle i want to work, but i cannot seem to gather the willpower to do so on a sunday evening to break out the pen and pad of paper, turn to section 3 of someone else's paper, and work out details to technical lemmas.

i'd much rather it be easy. anyone would.

experience tells me, however, that if i don't put in the work, then i won't really understand any of it .. at least to the degree that i can effectively use their results.

i'm not so tired from the week before and there is nothing stopping me from doing what i described above. on the other hand, i don't feel well-rested, either. there is the risk that

• i start working,
• everything's going well,
• and suddenly it's .. 2am? fvck!

my original intention would have been, as always, to get up bright and early at 7am [2] and make use of the pure productivity of the morning, as to make some significant progress on my own solo [3] projects ..

.. because i know, from experience, that the time i spend at the office will be productive, but not nearly as much as those first morning hours.



as for those footnotes ..

a day in the life: to-do lists? to-don't!

today i tried to be responsible,
so i tried to do as many of the items on my to-do list as i could.
        i can say with definiteness that it wasn't worth it.

maybe completing to-do lists just takes practice.
maybe i didn't take enough breaks,
maybe i didn't spend enough of the morning attempting creativity,
        brainstorming for research .. and just got frustrated at the lack of progress.

at any rate, running all these little errands has left me tired and unmotivated to ever do them again.

there's a reason, i suppose, why my to-do list is never fully crossed out and why i "procrastinate" with other things, like looking things up for research notes.

i guess i'm just spoiled. 7-:

mildly interesting: factoring numbers, self-similarly.

a friend of mine sent me the link to these animated factorisation diagrams, which are captured below as stills.

primes are displayed as circles ..

whereas for each factor, like the 5's in 250 or the 2's and 3's in 216, another level of  self-similarity is added.

admittedly i kept waitingfor another power of 2 or 3, just to see more iterations of four-corner sets and sierpinski-type triangles .. (-:

mildly relevant: crowd-sourced research, and could it work for maths?

when applying for grants, most of the time i expect a panel consisting of more senior researchers, possibly even a few of my peers.

this possibility, on the other hand, changes the crowd quite a bit!

Crowdsοurcing curiοsity-driven biοmedical research

Fact: the average basic-research life scientist deals with an 80% grant rejection rate, and gets his or her first big government grant at age 42. Basic biomedical research uses advanced 21st century technology, but is still fueled by a clumsy, archaic government-grant funding model that even predates the Internet.

It’s time scientists experimented with the way we all experiment.

Today, there’s a glut of highly trained but underemployed scientists. Let’s harness their idealistic passion before they turn grey, using social networks and data sharing to create an open, interactive, dynamic model of basic life sciences research. That new foundation can serve as a platform on which others will build and improve. This is particularly vital for mental health research, so often stymied by misunderstandings and blind spots, both public and scholarly.

// more @ rockethub.

to state the obvious, the difficulty is to find a way to show a project's significance to those who have the funds.

as for who might have the funds, likely they consist of normal, upper-middle class people and due to self-selection, more likely a tech-friendly professional crowd; who else, after all, would pay attention to this kind of proposal outside of their daily life?

despite a tech-friendly crowd, though, this wouldn't be easy. i doubt it would work at all for pure mathematics, and most of you (mathematicians) probably agree already. the point, however, is to figure out why, and a few reasons come to mind.

mildly relevant: amusing jargon ..

the terminology in metrιc space geοmetry is awesome;
i'm thinking of jargon like ..

• "prοpeller" (as in prοpeller cοnjecture)
  
• "exρander" (as in exρander graph)
• "Earthmοver" (as in transpοrtation cοst distances on a space of measures)
• "Tug-οf-War" (as in stοchastic games and ΡDEs)
• "Lamplighter" (as found in rigidity of quasi-isometries and metric distorsion)
• .. even "Pοlitics!" (as used in absοlute minimisιng Lipschιtz extensiοns)

mildly interesting: i don't see a singularity .. but still cool.

over at abstractstrategygames they've posted about a so-called singularly chess, where the pieces and rules are the same as usual (western) chess but the boards are non-linearly tiled ..

.. almost like a polar coordinate system;
as a result, pieces move in unconventional ways:

for more examples, visit the abstractstrategygames website (here).

i remember.

on a more sombre note, i just realised:
five years ago, the advisor passed away at the age of 47.
..
..
the thing is .. more years have now passed than the number of years that i've known him.  it's a very odd feeling, but maybe not:

the people that influence us most and the good mark they leave on our lives .. these are things that cannot really be quantified and should not be, either.

i've known some people all my life and others for just a few days or hours, and the time i've spent with each of them is hardly proportionate to their place in my life and thoughts.

so what are the rules of arithmatick, the measurement of time, in regards to people?

i still remember.  i still think of him,
less often than i did before ..

.. but i still do.

mildly relevant: for collaboration purposes?

interesting:

from mathιm

it would be more convenient, though, if they allowed rendering through LaTeX .. (-;.

memories: when i was a (mathematical) gun-for-hire ..

the season of job applications is in full swing, isn't it? [0]

in point of fact, last month i helped a colleague with a research statement [1], and a week ago my old officemate sent me his "research statement for non-experts" .. which seems apt, since his work is as far removed from mathematical analysis as i can think of.

every time i see a job ad that's not on mathjοbs or the usual channels, i pass it to a colleague who's on the market and seems to fit the bill.

there were some thoughts that i wanted to write about last year, but i was always too busy with this application, writing another talk, or chasing another research idea ..

.. more on the latter case, below.

maybe this year i can make up for that lost time .. and in the process, those of you on the market can have a welcome distraction from your own endeavours in the job market.

---

2010-11 was, in general, a bad year for me.

as an aside: rigorous conversation!

sometimes i think that i over-think casual conversation;

for instance, sometimes i forget and think about about how to respond to my friend's "how are you?" when (s)he simply meant hello. [1]

that said, yesterday i met an acquaintance, who is defending his Ph.D. today.  i paused for a second, after hearing the news, and then said:

"it could be that you believe in a completely deterministic universe;
but if not, then good luck!"

to his credit, he smiled briefly.

to be fair, one of my friends from school doesn't believe in luck .. so i never quite know how to wish her well in her endeavors ..


[1] it's not unlike looking at your watch when asked, "do you have the time?"
it is, after all, a yes/no question .. 7-:

from the arXiv: third derivatives could be useful ..

admittedly, i have always dismissed derivatives of orders 3 and higher.

maybe it's because my background in physics is poor, and i never rightly learned good mechanical interpretations of third-order derivatives.  (i still don't know how to think of them, honestly.)

at any rate, this title/abstract from the arXiv is suggestive.  maybe third derivatives are worth something, after all!

The Taylοr Expansiοn of the Expοnential Map and Geometric Applications

M. G. Mοnera, A. Montesinοs-Amilibia, E. Sanabria-Cοdesal

(Submitted on 22 Oct 2012)

In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in ${\bf R}^n$ up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in ${\bf R}^3$. Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in ${\bf R}^4$ and the asymptotic directions of a surface immersed in ${\bf R}^5$.

.